Testing the Combination of Belinostat and SGI-110 (Guadecitabine) or ASTX727 for the Treatment of Unresectable and Metastatic Conventional Chondrosarcoma

Inclusion Criteria: * Patients must have biopsy-proven conventional chondrosarcoma (CS) which is: * Either metastatic or locally advanced and unresectable, and * Measurable at study entry according to Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1 criteria, and * Amenable to biopsy with imaging guidance at no or acceptable risk to the patient as defined by institutional guidelines for research-related biopsies or the treating investigator's assessment * In addition, the following criteria must be met: * Patients must have at least one lesion measurable by RECIST version 1.1 criteria which has not been previously irradiated * Patients who have histologic evidence of grade 1 chondrosarcoma only must either be symptomatic from their disease in the opinion of the treating investigator or demonstrate radiographic evidence of disease progression in the 3 months prior to initiation of study treatment * Note: Pathology review and confirmation of diagnosis will occur at the site enrolling the patient on this study * Patients may have been treated with any number of prior systemic therapies. Because there are no Food and Drug Administration (FDA)-approved treatments for this disease, patients who have received no prior systemic therapy are also eligible. However, disease must be deemed surgically unresectable * Age \>= 18 years. Chondrosarcoma is rarely encountered in children and adolescents * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Absolute neutrophil count \>= 1,000/mm\^3 * Hemoglobin 8 g/dL * Platelet count \>= 75,000/mm\^3 * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2 x institutional ULN * Creatinine =\< 1.5 x institutional ULN OR glomerular filtration rate (GFR) \>= 45 mL/min/1.73 m\^2 * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression, if patients have been clinically asymptomatic, and if patients have not received systemic corticosteroids for at least 28 days. Patients with brain metastases not meeting these criteria are not eligible * Patients must be disease-free of prior invasive malignancies for \> 5 years, with the exception of curatively-treated basal cell or squamous cell carcinoma of the skin or carcinoma in situ of the cervix. * NOTE: If there is a history of prior malignancy, patients must not be receiving other specific treatment for that cancer * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better * The effects of belinostat and SGI-110 (guadecitabine) or ASTX727 on the developing human fetus are unknown. For this reason, and because the DNA methyltransferase inhibitor decitabine, the active metabolite of SGI-110 (guadecitabine) and a component of ASTX727, is known to be teratogenic, and because belinostat may cause teratogenicity and/or embryo-fetal lethality by virtue of targeting actively dividing cells, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation and for at least 6 months after the last dose of study drugs. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 3 months after completion of belinostat and SGI-110 (guadecitabine) or ASTX727 administration * Patients must be able to understand and willing to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible Exclusion Criteria: * Patients with dedifferentiated, mesenchymal, or clear cell chondrosarcoma are not eligible * Patients who have not recovered from adverse events (AEs) (i.e., have residual toxicities \> grade 1) due to prior anti-cancer therapy are not allowed, with the exceptions of alopecia and endocrinopathies from prior immunotherapy-based treatments that are well-controlled with hormone replacement. In addition, the following time periods must elapse between the last dose of prior anti-cancer treatment and initiation of study treatment on this protocol: * Cytotoxic chemotherapy or biologic, including immunotherapy: 28 days * Small molecule targeted drug: 21 days or 5 half-lives, whichever is shorter. If 5 half-lives is shorter than 21 days, then 21 days applies. * Radiation: 28 days, except for palliative radiation, for which 14 days applies * Patients who are receiving any other investigational agents * Patients with known history of allergic reactions or sensitivity attributed to compounds of similar chemical or biologic composition to SGI-110 (guadecitabine), its active metabolite decitabine, or ASTX727, or belinostat * Chronic use of any medications or substances that are strong inhibitors of UGT1A1 is not allowed. Patients must switch to alternative medications 7-14 days before treatment with belinostat. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * Patients with any known UGT1A1 polymorphism, heterozygous or homozygous, associated with reduced function (UGT1A1\*6, UGT1A1\*28, or UGT1A1\*60) * Patients with uncontrolled intercurrent illness * Patients with psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because SGI-110 (guadecitabine) is a derivative of decitabine, and ASTX727 contains the agent decitabine, which has the potential for teratogenic or abortifacient effects, and because belinostat may cause teratogenicity and/or embryo-fetal lethality by virtue of targeting actively dividing cells. Because there is an unknown but potential risk for AEs in nursing infants secondary to treatment of the mother with SGI-110 (guadecitabine), ASTX727 and belinostat, breastfeeding should be discontinued * Prolongation of the heart-rate corrected QT (QTc) interval \>= 450 ms (i.e., grade 1 or higher) on the screening electrocardiogram (ECG) prior to initiation of study treatment. If baseline QTc on screening ECG is \>= 450 ms (i.e., grade 1 or higher): * Check potassium and magnesium serum levels, and * Correct any identified hypokalemia and/or hypomagnesemia and repeat ECG to confirm a QTc interval \< 450 ms